Bottas survived a lunge from Vettel into Turn 1 at the subsequent safety auto restart and appeared to be heading for victory before a puncture handed Mercedes' first win of the year to Hamilton, with Kimi Raikkonen second and Sergio Perez beating Vettel to a shock third.
Hamilton finished comfortably ahead of team-mate Valtteri Bottas, while Daniel Ricciardo, who won the inaugural race at the Baku City Circuit a year ago, claimed fourth, an impressive result considering he only made it through to Q3 by a tenth of a second.
Red Bull Round 4 occurred on lap 35 and finally went the way of Ricciardo as he managed to pass his teammate around the outside at Turn 1. Vettel was running nine seconds clear of Valtteri Bottas before making his pit-stop on lap 30 for softs.
Ricciardo, who had previously passed Verstappen only to lose the spot during an exchange of pit stops, tried to slipstream his way around his teammate on the front straightaway, but when Verstappen weaved to block him, Ricciardo nosed into the back of Verstappen and both were sent crashing off the course in Turn 1.
The biggest setback for Red Bull saw the two team mates taking each other out.
Hamilton added: "It was really quite an emotional race to be honest".
The Australian, the victor last time out in China, did not back off and ended up smashing into Verstappen as he broke for the corner.
Bottas led the race and stayed out with the Red Bull drivers on the old tyres.
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They are five points ahead of third-placed Liverpool but 16 adrift of champions Manchester City with three games left. Nelson has 10 goals and 2 assists in 10 PL2 games this year at just 18 years of age.
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But nobody told me there is any decision. 'If I owned a club, I would think about him too. It is his decision and he is old enough to decide things like that.
It forced the Finn out of the race and handed the lead to Hamilton with just three laps remaining.
Ocon turned in for the apex with Raikkonen nearly completely alongside, and the ensuing contact pitched the Force India into the wall and out of the race and forced Raikkonen to pit for a new front wing. He dropped behind Hamilton and then Raikkonen at the next corner, before falling to fourth behind Force India's Sergio Perez on the following lap.
For Red Bull it was an afternoon that will require a stern debrief with their drivers.
On the restart, it was Bottas in front followed by Vettel, Hamilton, and Raikkonen.
The Safety Car was deployed again and the leaders pitted for new ultra-soft tyres to set up a charging 10-lap finale.
Ricciardo braked late and ran into the back of the younger man, who was attempting to cut back across and prevent his progress.
Canadian Lance Stroll, third in Baku previous year, will start 10th for Williams - the only team yet to score a point this season. On the cold tyres, as Grosjean weaved around behind the Safety Car to keep his tyres warm, he crashed into the barriers. Sainz was fifth after a strong race. Charles Leclerc, the rookie French driver in a Sauber, finished sixth and was voted the "Driver Of The Day".